In a saucepan cover potatoes with salted cold water by 1 inch and simmer, covered, until tender, 15 to 20 minutes. Drain potatoes in a colander and quarter. In a bowl stir together warm potatoes, oil, lemon juice, zest, chives, and salt and pepper to taste. Pit and peel avocado and cut into 3/4-inch pieces. Add avocado to potato mixture, gently tossing to combine.